Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| isoclinism | isoclinic class of D52 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#6036 |
| derived, Sylow | C52 ⋊ C22×C32×C4 |
| Hall | C32×C52 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D52 |
| commutator, hypocenter | C52 | C22×C32×C4 |
| Frattini | C2 | C2×C32×D52 |
| Fitting, socle | C22×C32×C52 | C22 |
| prime radical | C22×C32×C5⋊D5 | C2 |
| cosocle | C2×C52 | C23×C32 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C22×C32 | |
840 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 288 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 22 points:
(8,9)(10,11)(12,13)(14,15)(20,21,22);(1,2)(7,8,10,11,9)(17,18,19);(3,4,5,6)(13,14)(15,16)576 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C32×#87830, C2×C3⋊D3×#6036, C2×#72229304, … |
| maximal normal subgroup of | C2×C32×#87830, C2×C3⋊D3×#6036, C2×#72229304, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,9)(10,11)(12,13)(14,15)(20,21,22), (1,2)(7,8,10,11,9)(17,18,19), (3,4,5,6)(13,14)(15,16));